We discuss the lepton flavour and number violating conversion of bound
muons into electrons/positrons in detail. Given that experiments on
ordinary $\mu^-$-- $e^-$ conversion are expected to improve their
sensitivities by several orders of magnitude in the coming years, this
processes will lead to strong constraints on the model space which could
even challenge current LHC bounds. Furthermore, one can assume the limit
on $\mu^-$-- $e^+$ conversion to improve by roughly the same factor, which
means that this process might be a viable alternative to neutrinoless
double beta decay.
We introduce an effective field theory treatment for both processes,
respectively, to allow for a clean separation of nuclear physics from the
particle physics in order to make the respective computation accessible to
the particle physics community. For illustrative purposes, we discuss how
to tackle these computations by using a model in which the Standard Model
is extended by only a doubly charged scalar particle. We then illustrate
the potential impact of measuring muon conversion processes and point out
open issues.